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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 166635bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
166635.x1 | 166635bf1 | \([0, 0, 1, -101568, 38542014]\) | \(-1073741824/5325075\) | \(-574672312268556075\) | \([]\) | \(1824768\) | \(2.0921\) | \(\Gamma_0(N)\)-optimal |
166635.x2 | 166635bf2 | \([0, 0, 1, 898242, -943371387]\) | \(742692847616/3992296875\) | \(-430841345224016671875\) | \([]\) | \(5474304\) | \(2.6415\) |
Rank
sage: E.rank()
The elliptic curves in class 166635bf have rank \(1\).
Complex multiplication
The elliptic curves in class 166635bf do not have complex multiplication.Modular form 166635.2.a.bf
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.